I'm not entirely sure what you mean by "lift" the multiplication. Isn't multiplication on Z5XZ5 defined as (a,b)(c,d)=(ac,bd)?

No. Just like multiplication on RxR = C isn't defined by (a,b)(c,d) = (ac, bd) but by (a,c)(b,d) = (ac-bd, ad+bc).
Define by , where is the unique representative of degree less than 2 of an element in . This is a bijection. Then define , where the multiplication on the right is the one in . Then work out what this means in . Compare this to the formula for complex multiplication, and think about an explanation.